Finite Group Databasebeta

← #207010

Subgroups — #207010

Order 512.

862 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic
2#20536C211yesmaximal, characteristic
3#13964C211yesmaximal, characteristic
4#9889C212yesmaximal
5#9888C212yesmaximal
6#9887C214yesmaximal
7#8025C212yesmaximal
8#8013C212yesmaximal
9C2×#431C2212yes—
10#2857C2212yes—
11#2851C2212yes—
12#2627—22no—
13#2606—22no—
14#2605—22no—
15#2570C414yescomplemented
16#2550C414yescomplemented
17#2329—22no—
18#2283C2212yes—
19C2×#383C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×#376C2211yescharacteristic
22#1770—22no—
23#1770—22no—
24#1749C2212yes—
25#1749C2211yescharacteristic
26#1749C2211yescharacteristic
27#1441—22no—
28#1436—22no—
29C4⋊#95C2212yes—
30#1434C2212yes—
31#1294C2214yes—
32#1294C2214yes—
33#1294C2214yes—
34#1294C2214yes—
35C2×C4⋊Q8C2×C412yes—
36C2×#125C2×C412yes—
37C2×C22⋊Q8—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4D412yes—
40C22×C42—21no—
41C22×C42C2311yescharacteristic
42C22×C42C2311yescharacteristic
43C22×C42C2311yescharacteristic
44#431—24no—
45#431Q818yescomplemented
46C2×C4⋊C8—22no—
47C2×C22⋊C8C2312yes—
48C2×C22⋊C8C2312yes—
49C2×#97—22no—
50C2×#97—21no—
51C2×#97—21no—
52C2×C4×C8—22no—
53#400—24no—
54#398—24no—
55#398—24no—
56#395—24no—
57#391—22no—
58#391—24no—
59#389—22no—
60#389—24no—
61#389C2×C414yes—
62#389C2×C414yes—
63#387—24no—
64#387—24no—
65#387—24no—
66#385—24no—
67#384C2×C412yes—
68#383—22no—
69#383—24no—
70#382C2×C412yes—
71#382C2×C412yes—
72#382C2×C412yes—
73#381—22no—
74#380—22no—
75#380C2314yes—
76C4×C4⋊C4C2×C414yes—
77C4×C4⋊C4C2×C414yes—
78C4×C4⋊C4D412yes—
79C4×C4⋊C4D412yes—
80C4×C22⋊C4—22no—
81#376—22no—
82#376—24no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95D412yes—
90C2×#95C2312yes—
91C2×#95C2312yes—
92#336—22no—
93#336—21no—
94#336—21no—
95C22×Q8—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D412yes—
100C23×C4C2×D412yes—
101C23×C4C2×D411yescharacteristic
102C23×C4C2411yescharacteristic
103C22×C8—22no—
104C22×C8—22no—
105C4⋊Q8—82no—
106#125—82no—
107C22⋊Q8—44no—
108C22⋊Q8—44no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—82no—
120C2×C4⋊C4C22⋊C414yes—
121C2×C4⋊C4C22⋊C414yes—
122C2×C4⋊C4C22×C414yes—
123C2×C4⋊C4C22×C414yes—
124C2×C22⋊C4—24no—
125C2×C42—24no—
126C2×C42—44no—
127C2×C42—42no—
128C2×C42—81no—
129C2×C42—81no—
130C2×C42C22×C412yes—
131C2×C42C22×C412yes—
132C2×C42C22×C412yes—
133C2×C42C2×D412yes—
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D412yes—
137C2×C42C2×Q814yes—
138C2×C42#4114yes—
139C4⋊C8—28no—
140C22⋊C8—22no—
141C22⋊C8—22no—
142C22⋊C8C2×Q814yes—
143C22⋊C8C2×Q814yes—
144#97—82no—
145#97—81no—
146#97—81no—
147C4×C8—28no—
148#95—22no—
149#95—22no—
150#95—22no—
151#95—22no—
152#95—24no—
153#95—24no—
154#95—24no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—24no—
159#95—44no—
160#95—44no—
161#95—42no—
162#95—42no—
163#95—42no—
164#95—82no—
165#95C22⋊C414yes—
166#95C22⋊C414yes—
167C24C22×D411yescharacteristic
168C2×Q8—44no—
169C2×Q8—82no—
170C2×Q8—82no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4C2×C22⋊C414yes—
201C22×C4C2×C22⋊C412yes—
202C22×C4C2×C22⋊C412yes—
203C22×C4C2×C22⋊C412yes—
204C22×C4C2×C22⋊C412yes—
205C22×C4#12412yes—
206C22×C4#12412yes—
207C22×C4C4⋊Q812yes—
208C22×C4C4⋊Q812yes—
209C22×C4C23×C411yescharacteristic
210C22×C4C22×D411yescharacteristic
211C22×C4C22×Q812yes—
212C22×C4C2×#4112yes—
213C2×C8—24no—
214C2×C8—24no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—48no—
224C4⋊C4—44no—
225C4⋊C4—82no—
226C4⋊C4—84no—
227C4⋊C4—84no—
228C4⋊C4—82no—
229C4⋊C4—88no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C42—42no—
233C42—41no—
234C42—41no—
235C42—42no—
236C42—42no—
237C42—48no—
238C42—84no—
239C42C4×Q818yes—
240C42C22⋊Q814yes—
241C42C22⋊Q814yes—
242C23—21no—
243C23—21no—
244C23—22no—
245C23C22×C22⋊C411yescharacteristic
246C23C2×#12411yescharacteristic
247C23C2×C4⋊Q811yescharacteristic
248C23#57512yes—
249C23#57512yes—
250Q8—164no—
251C2×C4—24no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4#39114yes—
298C2×C4#39114yes—
299C2×C4C2×C4×Q812yes—
300C2×C4C2×C22⋊Q812yes—
301C8—44no—
302C8—44no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22C2×#39111yescharacteristic
312C22#288311yescharacteristic
313C22#288411yescharacteristic
314C22#289612yes—
315C22#290112yes—
316C22#313512yes—
317C22#316211yescharacteristic
318C22#316211yescharacteristic
319C4—24no—
320C4—24no—
321C4—22no—
322C4—22no—
323C4—44no—
324C4—42no—
325C4—42no—
326C4—84no—
327C4—84no—
328C4—84no—
329C4—162no—
330C2—22no—
331C2—21no—
332C2—21no—
333C2#1762312yes—
334C2#1762412yes—
335C2#2130711yescharacteristic
336C2#2132111yescharacteristic
337C2#2133411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
15. #2570C4semidirect
16. #2550C4semidirect
45. #431Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#351042
#205362
#139642
#98892
#98882
#98872
#80252
#80132